Originally Posted by

**jmarshall** Hello. I need to find the supremum and infimum of the set S={sqrt(n) - [n] : n in N the natural numbers}. [n] stands for the greatest integer m in Z such that m is less than or equal to n.

I assume the infimum is zero since you can't take the square root of a negative number (and fractions between 0 and 1 will be 0 in [n]) and I'm pretty sure there is no supremum since the function is unbounded. How would I prove these (both supremum and infimum) for an epsilon greater than zero?